Using set theory to reduce redundancy in pathway sets

The consolidation of pathway databases, such as KEGG, Reactome and ConsensusPathDB, has generated widespread biological interest, however the issue of pathway redundancy impedes the use of these consolidated datasets. Attempts to reduce this redundancy have focused on visualizing pathway overlap or merging pathways, but the resulting pathways may be of heterogeneous sizes and cover multiple biological functions.

R E S E A R C H A R T I C L E Open Access

Using set theory to reduce redundancy in

pathway sets

Ruth Alexandra Stoney1* , Jean-Marc Schwartz2 , David L Robertson2,3 and Goran Nenadic1,4

Abstract

Background: The consolidation of pathway databases, such as KEGG, Reactome and ConsensusPathDB, has generated widespread biological interest, however the issue of pathway redundancy impedes the use of these consolidated datasets Attempts to reduce this redundancy have focused on visualizing pathway overlap or merging pathways, but the resulting pathways may be of heterogeneous sizes and cover multiple biological functions Efforts have also been made to deal with redundancy in pathway data by consolidating enriched pathways into a number of clusters or concepts We present an alternative approach, which generates pathway subsets capable of covering all of genes presented within either pathway databases or enrichment results, generating substantial reductions in redundancy Results: We propose a method that uses set cover to reduce pathway redundancy, without merging pathways The proposed approach considers three objectives: removal of pathway redundancy, controlling pathway size and coverage of the gene set By applying set cover to the ConsensusPathDB dataset we were able to produce a reduced set of pathways, representing 100% of the genes in the original data set with 74% less redundancy,

or 95% of the genes with 88% less redundancy We also developed an algorithm to simplify enrichment data and applied it to a set of enriched osteoarthritis pathways, revealing that within the top ten pathways, five were redundant subsets of more enriched pathways Applying set cover to the enrichment results removed these redundant pathways allowing more informative pathways to take their place

Conclusion: Our method provides an alternative approach for handling pathway redundancy, while ensuring that the pathways are of homogeneous size and gene coverage is maximised Pathways are not altered from their original form, allowing biological knowledge regarding the data set to be directly applicable We demonstrate the ability of the algorithms to prioritise redundancy reduction, pathway size control or gene set coverage The application

of set cover to pathway enrichment results produces an optimised summary of the pathways that best represent the differentially regulated gene set

Keywords: Set cover, Data redundancy, Pathways, Gene enrichment analysis

Background

Pathways are sets of genes corresponding to functionally

related interacting proteins Pathway data is available

from many databases dependent on biological focus The

fragmented nature of pathways across multiple databases

makes it difficult to perform inclusive analysis of all

known data To address this issue, many attempts have

been made to consolidate pathway databases such as

ConsensusPathDB (CPDB) [1], PathwayCommons [2],

The Human Pathway Database (HPD) [3], Pathway

Interaction Database (PID) [4], and NCBI Biosystems [5] Amalgamating multiple databases into a consistent searchable format facilitates the use of these resources, however the arbitrary nature of pathway boundaries results in overlap, generating data redundancy This redundancy greatly increases the quantity and complex-ity of pathway data, which has lead to the development

of a range of tools to assist in data simplification and in-terpretation [3, 4, 6–8] Previous solutions presented to deal with redundancy include visualizing redundancy be-tween pathways to the user [3], merging pathways based

on similarity [7,8] and even integrating full pathway sets into a non-redundant, single unified pathway [9] Redu-cing redundancy simplifies the pathway-related descriptive

* Correspondence: ruth.stoney@manchester.ac.uk

1 School of Computer Science, University of Manchester, Manchester M13

9PL, UK

Full list of author information is available at the end of the article

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space, allowing multiple resources to be combined while

limiting the number of pathway attributes assigned to

each gene The advantages are apparent, with resources

such as PathCards being integrated into the widely used

GeneCards [8]

Redundancy Control in Pathway Databases (ReCiPa)

[7] uses a pathway merging algorithm to combine

path-ways with high levels of overlap Users select a

max-imum overlap threshold and pathway pairs displaying

greater levels of overlap are merged Within that study

redundancy was observed within five large databases

(KEGG, Biocarta, CGP, NCI-PID, and Reactome) They

proceeded to merge pathways from the Molecular

Signa-tures Database (MSigDB), whose overlap exceeded 75%,

reducing pathway redundancy

Pathcards described a multistep procedure to reduce

pathway redundancy, also through pathway merging [8]

Two thresholds were calculated and sequential merging

steps were used to minimize overlap, while preventing

the generated super-pathways from becoming too large

to be informative By merging pathways into

super-pathways, Pathcards suggested many new molecular

in-teractions They demonstrated that many of these newly

generated interactions are supported by high numbers of

literature co-mentions and high experimental

interac-tions scores according to STRING However, while the

generation of potential interactions can be highly

benefi-cial, if the aim is to utilize previously validated data,

merging pathways introduces a source of uncertainly

into the dataset

A major application of pathway data sets is pathway

explored the capability of their reduced pathway dataset

to improve enrichment results Enrichment analysis of

830 differential expression sets was performed using the

super-pathways generated within Pathcards The

enrich-ment results from super-pathways tended to be more

significant than the enrichment scores of their

constitu-ent pathways Similarly within the ReCiPa study

enrich-ment analysis was performed using genes differentially

expressed in obesity After merging, the top 20 most

significantly enriched pathways showed less overlap and

greater significance towards the disease, compared to

the original dataset

Pathway Distiller implemented an alternative approach

by removing redundancy from enriched pathway sets

following enrichment analysis [6] Pathways may be

consolidated into pathway concepts based on gene

expression profiles, gene membership, protein-protein

interaction data or shared Gene Ontology (GO) terms

Each method provides varying, complementary views of

the data, with different pathway concepts generated

Consolidating enrichment output into a reduced number

of pathway concepts increases data manageability and

readability, by organizing redundant pathways into their major groups

All of the approaches discussed to this point have used merging and consolidation to address redundancy Alexa

et al (2006) demonstrated that redundancy in GO en-richment results could be reduced by selecting a subset

of representative terms [10] Pathway enrichment ana-lysis and GO enrichment anaana-lysis are similar techniques

in which sets of differentially expressed genes are com-pared to gene sets associated with pathways or GO terms Alexa et al (2006) introduced two algorithms, elim and weight, which use the Gene Ontology topology

to select a representative subset of highly enriched GO terms [10] The enrichment set cover algorithm pre-sented in this paper shares some conceptual similarity with this approach however, the implementation is dif-ferent since there is no organized topological hierarchy for combined pathway datasets and the rules governing the Gene Ontology, such as the true path rule [11], do not apply

Within this paper we show that set cover can be used

to reducing redundancy by selecting subsets of represen-tative pathways We describe a set of algorithms for reducing redundancy in pathway datasets, as well as a separate algorithm for reducing redundancy from path-way enrichment results The proportional set cover algo-rithm and hitting set cover algoalgo-rithm aim to identify a minimum subset of pathways required to cover the genes in highly redundant, consolidated pathway data-bases The generated set covers are not designed to depict the full range of possible pathway boundaries and their accompanying cellular functions, but rather they provide a simplified set of pathways to represent the actions of genes within the dataset Since the pathways are not merged database and biological information remains directly applicable and functional specificity is not lost through pathway size expansion The proposed method also removes the risk of biologically distinct pathways being merged The algorithm’s ability to re-move overlap is not limited by thresholds, conferring an advantage compared to approaches such as Pathcards and ReCiPa in which redundancy between pathway pairs can only be removed if the overlap exceeds the thresh-old Set cover algorithms also consider redundancy be-tween multiple pathways, rather than just comparing pathway pairs

We also developed the enrichment set cover algorithm for handling pathway enrichment data and applied it to

a set of enriched osteoarthritis pathways [12] In contrast

to the approaches used by ReCiPa and Pathcards, the enrichment set cover algorithm is designed to be used

performed using the full pathway dataset Redundancy is then removed from the enriched pathway set by

selecting the pathway with the lowest p-value to cover

each differentially regulated gene Enriched pathways are

not merged or altered and the number of enriched

path-ways required to cover the dataset is reduced The

resulting pathways set can therefore be used as an

opti-mized summary output, conveniently showing the most

important pathways for describing the differentially

reg-ulated gene set By increasing the number of

differen-tially regulated genes covered by the most highly

enriched pathways, researchers examining the top 10 or

20 pathways are provided with a more inclusive

por-trayal of the gene set

Approach

We downloaded pathway data from ConsensusPathDB

(CPDB), an opensource online collection of pathways,

that incorporates 32 sources including KEGG,

Wikipath-ways, PDB, Reactome CPDB makes these resources

available as a single download, which we acquired on

24/09/2015 containing 4011 pathways We applied the

set cover algorithm to the CPDB data set, analyzing it’s

effectiveness at: reducing pathway overlap; reducing

pathway size variability; and preserving the maximum

number of genes in the data set We found that standard

set cover caused unacceptable increases in pathway size,

therefore we modified the algorithm and assessed the

modified algorithms capability to meet the previous

three objectives

Set cover is a well-defined algorithm in computer

sci-ence for handling overlapping sets of sets For example,

set cover is used by CLASS, a bioinformatics program

that maps RNA sequence data to transcripts [13] Set

cover has also been used to predict protein-protein

interactions based on binding domains [14], to reduce

the complexity of SNP sets [15] and to minimize the

number of probes needed to analyze DNA [16]

Set cover algorithms deal with elements and sets,

which relate to genes and pathways respectively All the

unique genes in the data set are collectively referred to

as the universe The aim is to produce a reduced

selec-tion of sets (pathways), which collectively cover all the

elements (genes) in the universe (dataset) This subset of

the original data is called the cover set [17] Each time a

pathway is added to the cover set the genes in the

path-way become covered (Fig 1) Direct application of set

cover lead to extremely large, functionally non-specific

pathways dominating the cover set, therefore we

imple-mented the proportional set cover and hitting set cover

algorithms to better control pathway size, while reducing

redundancy and covering the dataset

When dealing with enrichment analysis data the aim is

to reduce redundancy between pathways, while

preser-ving the order of enrichment significance denoted by the

p-values We designed an algorithm that would select

the set of pathways with the lowest p-values capable of covering all the genes in the dataset This ensures that the filtered results return the most enriched pathways available for each gene

Methods

Overlap score

To measure overlap across different algorithms we mea-sured the mean number of pathways that each gene ap-pears in Within the raw data genes appeared in a mean

of 12.4 pathways We refer to this metric as the overlap score

Set cover

We applied the set cover algorithm to the data set, which generates a subset of pathways called a cover set,

in which all the genes in the data set are represented or

“covered” Set cover begins by first assigning values to each pathway (vi) The set cover values correspond to the number of uncovered genes each pathway contains (Eq.1)

where (si) is the pathway’s gene set and R is the set of all uncovered genes

At the beginning of the algorithm all the genes in the dataset are uncovered so the algorithm selects the lar-gest pathway The genes from the selected pathway are then covered, so it is unnecessary to cover them again using additional pathways The algorithm then recalcu-lates how many uncovered genes each pathway contains and continues to add the pathway with the maximum value to the set cover until all genes in the data set are covered

whereR is the set of uncovered genes, U is all the genes

in the dataset,C is the covered genes, SC is the set cover result, GC is the gene coverage (see Gene Set Coverage section) andsiis a pathway

Application of the set cover algorithm was effective in reducing overlap between the pathways; however, it se-lected very large pathways with reduced informativeness

(maximum size 2320, standard deviation 160, almost

double the standard deviation on the original dataset

86.9) We therefore explored methods that avoid

prefer-ential selection of large pathways

Gene set coverage

As the set cover algorithm approaches completion and

the final sets are added to the cover set, increases in

data coverage are gained at the expense of redundancy

reduction This is because the final sets required to

cover the few remaining genes tend to have the most

overlap with other pathways already in the set cover In

addition, fewer pathways are available to cover the final

few genes, restricting options to control pathway size

To allow a user-defined compromise between the gene coverage, pathway redundancy and pathway size we introduce the Gene Coverage (GC) parameter Setting

GC below 100% allows the algorithm to finish before the final elements have been covered We experimented setting GC to 90, 95, 99 and 100% of the number of genes in the data set

Proportional set cover

When reducing pathway redundancy there are three competing aims: reducing redundancy; controlling path-way size; and covering the entire gene set The propor-tional set cover algorithm was generated to focus on controlling pathway size

Fig 1 Set cover a A simple set of overlapping sets b The red set with 8 uncovered elements is selected first c The blue set with 3 elements is selected second d The orange set then covers all the elements in the universe

To control the size of the pathways we altered the

scoring mechanism to rank pathways based on the

pro-portion of uncovered genes they contained, rather than

the absolute number (Eq 2) This works because larger

pathways are more likely to have a proportion of their

genes covered when other pathways are selected

Add-itionally this mechanism directly penalizes overlap,

which the standard algorithm does not At the beginning

of the proportional set cover algorithm none of the

genes are covered so the proportion of uncovered genes

in every pathway is 1 This would result in the starting

pathway being selected at random To ensure that

path-way size variability is controlled as strictly as possible,

we implemented the second part of Eq.2, which ensures

that pathways of mean pathway size are preferentially

selected when multiple pathways with the same

propor-tion of uncovered genes are available

vi¼j sj si∩R j

ij þ

1

wheresiis the pathway’s gene set, jsijis the mean

path-way length,R is the uncovered genes set and k is a large

constant to limit the influence of the second term (taken

equal to 10,000)

Hitting set cover

The set-covering problem can be reformulated into the

equivalent set-hitting problem In this formulation genes

and pathways are visualized as bi-partite graph in which

the pathways are connected to the genes that they

con-tain In this depiction it is clear that some genes are only

linked to a single pathway, which must be selected if the

gene is to be covered The importance of pathways can

therefore be considered as a factor of how infrequent

their genes are The hitting set cover is therefore

designed to reduce redundancy as much as possible

without directly selecting for pathway size

We counted instances of each gene j within the

path-ways of the data set (fj), then assigned the gene’s value vj

as 1/ fj(Eq.3) We then assigned a value vito each

path-way defined as the sum of each uncovered gene’s scores

divided by the number of genes in the pathway (Eq.4)

vi¼

P

jϵs i ∩ Rvj

where vjis the value of a gene, fjis the number of

path-ways the gene is in,

jϵsi∩ R means for each uncovered gene in the pathway

and |si| is the size of the pathway

Set cover for pathway enrichment analysis

Pathway analysis is a frequently used method; therefore

a modified set cover algorithm to address this situation could be highly useful The universe represents differen-tially expressed genes and the sets are enriched pathways generated through enrichment analysis Enrichment ana-lysis results represent entirely different input data com-pared to the pathway datasets used in the previous algorithms, as the enriched pathways already have scores (p-values) We wish to reduce redundancy (gene overlap) between enriched pathways and it is essential that the pathways with the lowest possible p-values are selected

Eq.4allows the pathways with the lowest p-values to be selected, unless all of their genes are covered by other enriched pathways with even lower p-values

wheresiis the enriched pathway’s gene set, R is the un-covered gene set,∅ is an empty set, b is a binomial op-erator, pvaluei is the pathway’s p-value and vi is the pathway’s set cover value

We generated the enriched data set by applying GOseq [18] to expression data from the damaged cartil-age in osteoarthritis patients and controls [12]

Results

We started with the large, extensively redundant CPDB data set and used set cover to reduce pathway overlap, while controlling pathway size and seeking to cover as much of the data set as possible We describe the ability

of the standard set cover algorithm and two modified algorithms, in conjunction with the GC parameter, to meet these objectives

Remaining pathway redundancies vary depending on the applied algorithm

The original pathway data set contained 11,196 genes and

3305 pathways; the starting overlap score (see Methods) was 12.4 The standard set cover algorithm reduced over-all redundancy from 12.4 to 4.1, a 73% reduction (since a completely discrete pathway set would have a score of 1) The overlap score for proportional set cover was 4.36, slightly higher than the standard set cover algorithm, but still representing a 70% reduction in overlap from the original data The hitting set cover algorithm was designed

to select pathways that contained rare genes within the data set, resulting in the greatest reduction in overlap (overlap score of 3.95 equivalent to a 74% reduction, see Overlap scoresection for calculation)

After application of the set cover algorithms the dis-tribution of the remaining overlap between pathways varied greatly Figure 2 shows the Jaccard similarity

Fig 2 Jaccard coefficient between pathway pairs in the cover set results produced by each algorithm

Fig 3 Redundancy in set cover outputs given different GC values

between pairs of pathways, in the outputs produced by

each of the three algorithms We used the Jaccard

simi-larity to measure the degree of overlap between

individ-ual pathway pairs This score measures the proportion

of genes shared across both pathways The standard set

cover algorithm produced the lowest maximum overlap

(Jaccard similarity = 0.68) between the pathways in the

output dataset However, compared to the original data,

a higher proportion of pathway pairs in the set cover

output showed Jaccard similarities between 0.1 and 0.3

Proportional set cover had the greatest maximum

Jaccard similarity at 0.93, out of the set cover algorithms

The hitting set cover algorithm produced a maximum

Jaccard similarity between two pathways of 0.82, despite

having the lowest overlap score

Gene coverage can be lowered to reduce redundancy

For each of the algorithms it is possible to use the GC

parameter to prioritize reductions in redundancy over

gene coverage by stopping any algorithm before all of

the genes in the dataset have been covered Figure 3 shows improved ability of the set cover algorithms to re-duce pathway overlap for different values of GC If 99%

of the genes are required then the hitting set algorithm achieves the lowest overlap score of 3.24, equivalent to

an 80% reduction in overlap Redundancy can be further reduced if only 95% of the genes are covered, with the proportional and hitting set algorithms producing an overlap score of 2.41, equivalent to a 88% reduction in redundancy Both the proportional set cover and the hitting set cover are more effective at reducing redun-dancy than the standard set cover if GC is set to less than 100%

Pathway size is affected by the set cover algorithm and gene coverage setting

When GC was set to 100% the standard set cover algo-rithm represented all of the genes in the dataset using only 524 pathways (16% of the original pathway set) However, many of these were very large increasing the

Fig 4 Pathway sizes in cover set when GC is set to (a) 100%, b 99%, c 95% and (d) 90% The boxes indicate the 25th and 75th percentiles and the whiskers indicate the 5th and 95th percentiles

mean size to 87.2 (standard deviation 160.1) These

pathways have reduced informativeness since functional

specificity is lost Figure 4a illustrates the tendency of

this algorithm to select extremely large pathways

The proportional set cover algorithm was designed to

preferentially select moderately sized pathways This

returned a cover set of 1336 pathways with controlled

size variation (mean of 36.5, standard deviation 55.1)

shown in Fig 4a The hitting set cover algorithm was

less able to control pathway size than the proportional

set cover algorithm, returning 957 pathways with a mean

size of 46.2 (standard deviation 61.7)

Figure 4b–d show that as GC is reduced the tendency

of the standard set cover to select very large pathways

be-comes more exaggerated Decreasing GC also improves

the ability of the proportional set cover algorithm to select

moderately sized pathways The hitting set algorithm also

tends to select smaller pathways when GC is reduced,

since larger pathways often contain more frequent genes

Reducing GC affects pathway size since in the later stages

of the algorithm, fewer pathways are available to cover the

remaining genes, reducing the available options

There-fore, lowering GC has the ability to help control pathway

size when the proportional set cover and hitting set cover

algorithms are used

Since the databases that contribute to CPDB contain

pathways of different sizes, the set cover generated may

preferentially select pathways from some databases more

than others

Table 1 shows the proportion of pathways that come

from each database in the cover set generated by each All

algorithms generate set covers with reduced INOH and

SMPDB pathways, showing that SMPBD’s focus on small

molecules and INOH’s ontology-based approach tend to

be ill-suited to the generation of discrete pathway protein sets The standard set cover algorithm generates sets con-taining large pathways, preferentially selecting pathways from KEGG (median size 65, see Table 1) and Netpath (median size 51); while proportional set cover tends to select smaller pathways from Reactome (median size 17) and HumanCyc (median size 5), whilst avoiding NetPath

Reducing redundancy in pathway enrichment analysis

To demonstrate the ability of the set cover algorithm to handle enrichment data, we applied the enrichment set cover algorithm to an osteoarthritis data set, retrieved from Dunn et al (2016) [12] From the osteoarthritis data set, 58.3% of the differentially expressed genes could be mapped to a CPDB pathway, which was a 17% improvement on the GOseq [18] implemented data set

We retrieved 42 enriched pathways with a p-value lower than 0.05, following the Benjamini-Hochberg correction for multiple testing Set cover for enrichment analysis reduced the number of pathways required to cover the differentially expressed genes to 23 (Additional file 1: Table S1)

The heat map in Fig.5ashows the asymmetric overlap between the top ten pathways before application of the algorithm The p-values from pathway enrichment deter-mine the order in which pathways were considered for inclusion in the cover set Pathways were omitted if all

of the differentially expressed genes that they covered were also covered by more enriched pathways Note that overlap tends to be higher in the bottom left triangle as pathways added later were often smaller subcomponents

Table 1 Proportion of pathways from CPDB databases

Median

size

CPDB

% Standard set cover Hitting set cover Proportional set cover

Reactome 17.0 39.6 4.2 5.3 10.8 21.2 36.1 35.1 34.7 35.3 39.4 40.9 45.1 48.8

Wikipathways 26.0 8.8 25.6 24.9 28.9 25.3 15.6 16.0 16.2 16.2 14.2 13.7 12.5 11.7

Median size represents the median sizes of the pathways in the CPDB dataset CPDB % represents the proportion of the pathways in the unaltered dataset that came from each database The following columns represent the proportion of pathways in the set cover generated by the standard set cover algorithm, the hitting set cover algorithm and the proportional set cover algorithm Different results are obtained by altering the proportion of the gene set covered, shown in

of larger pathways We can see that ‘extracellular matrix

organization’, the most enriched pathway, was placed in the

cover set first Next was‘collagen biosynthesis and

modify-ing enzymes’; however, all of the differentially expressed

genes in this pathway are also covered by the larger

path-way ‘extracellular matrix organization’, as indicated by the

red cell in the ‘collagen biosynthesis and modifying

enzymes’ row, ‘extracellular matrix organization’ column

The corresponding cell in the ‘extracellular matrix

organization’ row reveals that 24% of the differentially

expressed genes in ‘extracellular matrix organization’ are

also in‘collagen biosynthesis and modifying enzymes’

Figure5bshows overlap between the top ten pathways after application of the enrichment set cover algorithm Because the differentially expressed genes covered by the

‘collagen biosynthesis and modifying enzymes’ pathway are a subset of those covered by the‘extracellular matrix organization’ pathway, the ‘collagen biosynthesis and modifying enzymes’ pathway is removed from the cover set (Fig 5b) The second pathway in this list therefore becomes ‘GPCR signaling g alpha q’ The ‘collagen for-mation’ and ‘class b 2 secretin family receptors’ pathways are also removed because the differentially expressed genes they cover are additionally covered by the more

Fig 5 Pathway redundancy heat maps a Pathway overlap for top ten enriched pathways b Pathway overlap for top ten enriched pathways after application of set cover The values represent asymmetric overlap, i.e for each pathway shown on the left axis, values represent the proportion of genes that are also included in the pathway shown on the bottom axis

enriched pathways ‘extracellular matrix organization’

Additionally,‘GPCR signaling pertussis toxin’ and ‘GPCR

signaling cholera toxin’ are absent from the returned list,

as all of their differentially expressed genes are found in

‘GPCR signaling g alpha q’ or ‘signal transduction’

Some pathways in the enrichment set cover do still

show high levels of overlap, for example ‘wnt signalling

network’ is included despite 89% of its differentially

expressed genes being covered by ‘signal transduction’

This is acceptable because ‘signal transduction’ is more

highly enriched than ‘wnt signalling network’, yet the

‘wnt signalling network’ is worth including as it contains

three differentially expressed genes that are not in‘signal

transduction’ If ‘wnt signalling network’ had been

excluded then these genes would not have been

described by the most significant pathway available to

represent them The unmodified top ten enriched

path-ways only cover 78.0% of the enriched genes Using the

set cover enrichment algorithm increases this figure to

85.2% without disrupting the pathway order given by the

enrichment p-values

Discussion and conclusion

We described algorithms suitable for reducing overlap in

large pathway data sets allowing multiple databases to

be amalgamated without excessive redundancy impeding

the usefulness of the resource Standard set cover is the

best algorithm to reduce the number of pathways

required to cover the data set, but significantly increases

pathway size, which can be controlled by proportional

set cover or hitting set cover The proportional set cover

is the best algorithm for controlling pathway size and

the hitting set cover is the preferred choice for covering

all of the genes in the dataset with minimal pathway

redundancy We showed that reducing the GC

param-eter allows further reductions in pathway redundancy;

for example, if only 95% of the genes in the CPDB

data-set were covered redundancy can be reduced by up to

88% In addition reducing GC increases pathway size

control when the proportional set cover and hitting set

cover algorithms are used

For pathway enrichment analysis we aimed to

re-duce redundancy while selecting the most significantly

enriched pathways based on p-values As an

applica-tion we used the modified set cover algorithm to

re-duce the results of enrichment analysis from a large

osteoarthritis data set We found that 5 out of the 10

top ranking pathways could be omitted as they were

subsets of more highly enriched pathways Overlap

between pathways returned from enrichment data is

not always immediately obvious and requires further

consideration By reducing this redundancy, data

redundancy also allows the user to explore substan-tially more of the data set using the same number of pathways

The enrichment set cover algorithm presented within this study differs from existing methods implemented by ReCiPa and Pathcards, since enrichment analysis is performed prior to reduction of redundancy This is be-cause the different sets of pathway boundaries available

in the full dataset may optimally fit the differentially expressed genes For example, comparison of the ‘apop-tosis’ taken from KEGG, Reactome and Wikipathways, reveals that many of the proteins are specific to a single database [19] This is due to the vague definition of pathway boundaries, as well as differing experimental focus on cellular contexts, such as tissues or disease states Following enrichment analysis the pathways that are most significantly enriched are selected to represent the differentially expressed genes and superfluous path-ways are removed This prevents the top results from being dominated by large numbers of highly similar pathways

Set cover uses greedy heuristic methods, which pro-vide good approximations of the optimal solution in a time effective manner These methods are extremely effi-cient and can be run in a matter of minutes, however it should be noted that they do not guarantee an optimal solution This is particularly true for the proportional set cover algorithm where the randomness of early selec-tions influences the result However, all possible out-comes result in reduced redundancy The enrichment set cover algorithm is exempt from these considerations unless multiple pathways have identical p-values

It should also be noted that set cover algorithms, in-cluding the algorithms presented within this paper, are capable of reducing redundancy regardless of structure

of pathway overlap within the initial data set [20, 21] That is to say, the methods presented are equally appro-priate for use on data sets in which many pathway pairs show high levels of overlap or datasets with low overall overlap The selection criteria used within each approach (the number of uncovered genes; the highest proportion

of uncovered genes; the rarity of genes across the data-set; and enrichment p-values) utilize relative measure-ments taken across the dataset and are not affected by the overall composition of pathway redundancy The only assumptions made are that some variability will be present in both the size of pathways and the redundancy between pathway pairs Since pathway datasets are un-likely to be entirely uniform in each of these aspects, the methods described are capable of iteratively selecting pathways on the basis of their relative viscores across all foreseeable datasets

We have provided a method to dramatically reduce re-dundancy in pathways facilitating a more concise